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user21820

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About

Foundations

  • Axioms & Definitions
  • Contradiction & Game semantics & True induction
  • Circularity in mathematics & 'Paradoxes' resolved
  • Building blocks & Incompleteness theorems
  • Logic texts

Natural deduction

  • Rules & Examples

Rigour + Intuition

  • Naturals
  • Reals & Real exponentiation & $\cos,\sin$ & $\exp(iπ)+1=0$
  • General differentiation & applications
  • Matrix multiplication

Beautiful stuff

Puzzles:

  • Rubik's cube & other permutation puzzles

Finite difference:

  • Indefinite summation (e.g. sum of cubes)
  • Linear recursive sequences & their indefinite sum

Asymptotic expansion:

  • k-times-differentiable function
  • Central difference formulae
  • Newton-Raphson convergence criteria
  • Systematic computation of limits

Smoothing:

  • To prove AM-GM & Jensen's inequality
  • Maximum area of polygon given perimeter
  • Maximum expression value using brackets

Multi-dimensional analysis:

  • Multi-dimensional IVT, which proves:
  • Poincaré–Miranda theorem
  • Ham sandwich theorem [consider all hyperplanes that bisect one piece]

Inversion:

  • Apollonius' Problem

Common oversights

  • Separating variables
  • 'One-step' analysis to find expectation
  • Explanation of induction

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